Prediction of steady state input multiplicities for the reactive flash separation using reactioninvariant composition variables

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1 Insttuto Tecnologco de Aguascalentes From the SelectedWorks of Adran Bonlla-Petrcolet 2 Predcton of steady state nput multplctes for the reactve flash separaton usng reactonnvarant composton varables Jose Enrque Jame-eal, Unversdad de Guanajuato Adran Bonlla-Petrcolet Juan Gabrel Segova-Hernandez, Unversdad de Guanajuato Salvador Hernandez, Unversdad de Guanajuato Avalable at:

2 2 th European Symposum on Computer Aded Process Engneerng ESCAPE2 S. Perucc and G. Buzz Ferrars (Edtors 2 Elsever B.. All rghts reserved. 685 Predcton of Steady State Input Multplctes for the Reactve Flash Separaton usng Reacton- Invarant Composton arables José Enrque Jame-eal a, Adrán Bonlla-Petrcolet b, Juan Gabrel Segova- Hernández a and Salvador Hernández a a Unversdad de Guanajuato, 365, Méxco, gsegova@ugto.mx b Insttuto Tecnológco de Aguascalentes, 2256, Méxco, petrcolet@hotmal.com Abstract In ths study, we report a new approach for predctng and analyzng steady state nput multplctes n reactve flash separaton processes. ecessary condtons are suggested to detect these multple steady states usng reacton-nvarant composton varables. Our results suggest that the condtons proposed for dentfyng the presence of nput multplctes of non-reactve systems are useful for the analyss of reactve systems f transformed composton varables are employed. Performance of our approach s llustrated usng MTBE producton as case of study. Keywords: Reactve flash separaton, multple steady states, MTBE producton. Introducton Input multplcty s an mportant feature of ndustral processes and plays an mportant role n desgn, smulaton and control of separaton unts []. In process system engneerng, t s mportant to predct all multple steady states wthn the practcal doman of operatng varables, to know whether they are desrable, and to understand how the separaton scheme responses to changes n operaton condtons. To date, theoretcal and expermental studes have shown the exstence of multple steady-states (MSS n both non-reactve and reactve separaton processes. In partcular, due to the nteracton between chemcal reacton and phase equlbrum, reactve separaton processes exhbt hgh non-lnear behavor and MSS solutons are often possble. The exstence and possble explanaton of multplcty n reactve separaton processes for specfc systems (e.g., MTBE, ETBE or TAME producton have been nvestgated by several authors especally n reactve dstllaton columns [2]. However, only few studes have been conducted on multplcty n reactve flash separatons. Despte sgnfcant progress on desgn and modelng of reactve dstllaton columns have been acheved, more general results are needed concernng the steady state multplcty n reactve flash operaton. It s convenent to remark that the determnaton and analyss of the exstence of MSS n reactve separaton processes stll appears to be a challengng task and t s more complex than those nvolved for conventonal separaton schemes. Based on ths context, n ths study we provde new condtons for the exstence of MSS n reactve flash separatons. These condtons are based on the applcaton of reactonnvarant composton varables of Ung and Doherty [3]. The presence of the nput multplcty s studed for MTBE producton as example of our approach. For ths reactve system, we report the exstence of MSS for dfferent operatng condtons and we show that necessary condtons for the exstence of multple solutons n nonreactve flash operaton, prevously reported by Monroy-operena [] and Tscareño et

3 686 J.E. Jame-eal et al. al. [4], can be extended for reactve systems f reacton-nvarant composton varables are used n multplcty analyss. 2. Multplcty analyss of flash separaton n reactve systems Gven a temperature T and pressure P, consder a flash separaton process for a system of c components that undergoes r ndependent chemcal reactons. The thermodynamc problem for modelng the reactve vapor-lqud equlbrum s to solve the non-lnear equaton system that nvolves the materal balance, phase and chemcal equlbrum, and consstency equatons. Usually, ths problem s formulated usng conventonal composton varables (.e. mole numbers as ndependent varables and unknowns. However, as ndcated by Ung and Doherty [3], the numbers of moles are not the natural composton varables to use n the modelng of reactve systems because they do not have the same dmensonalty as the number of degrees of freedom (.e. they are nconsstent wth respect to the Gbbs phase rule. To obtan a convenent descrpton of reactve vapor-lqud equlbrum problem and to smplfy the analyss of MSS n flash systems, we have appled the reacton-nvarant composton varables proposed by Ung and Doherty [3]. These varables are based on transformaton of the physcal compostons and ts prncpal beneft s that the chemcal and physcal equlbrum approach n the reactve mxture s dentcal to a strctly physcal equlbrum model. The dmenson of reacton-nvarant composton space s equal to the number of degrees of freedom obtaned from the Gbbs phase rule. Thus, these varables depend only on the ntal composton of each ndependent chemcal speces, restrct the soluton space to the compostons that satsfy stochometry requrements and also reduce the dmenson of the composton space by the number of ndependent reactons. These features allow all of the procedures used to model non-reactve mxtures to be extended to systems subject to chemcal equlbrum and, consequently, non-reactve algorthms can be easly modfed to account for chemcal reactons [3]. Therefore, transformed mole varables for reactve systems are defned as = ( x v ( v for =,..., c r ( where x s the mole fracton of component, x ref s a column vector of mole fractons for r reference components, v s the row vector of stochometrc coeffcents of component for each of the r reactons, s an nvertble and square matrx formed from the stochometrc coeffcents of the reference components n the r reactons, and v s a row vector where each element corresponds to the sum of stochometrc coeffcents for all components that partcpate n each of the r reactons, respectvely. Transformed mole fractons are related to conventonal mole fractons x usng the reacton equlbrum constant K eq. It s mportant to note that the set of has the desrable property of takng the same numercal values before and after the reactons. Ths s n contrast to conventonal mole varables x and n, whch have dfferent values for the components n the unmxed and mxed (.e., reactng states [3]. Based on ths fact, the reactve vapor-lqud equlbrum problem s modeled usng as follows [5] where = ( Z δ ˆ φ ( ˆ φ ( K θ + for =,..., c r (2 = K θ + δ = ( Z K θ + δ ( ˆ φ ( ˆ φ ( K θ + for =,..., c r (3

4 Predcton of steady-state nput multplctes for reactve flash separaton 687 θ = ( v δ = ( v ref ( K x ( v x ref ( v x ref (4 subject to the materal balance Z ( ˆ φ ˆ φ = for =,..., c r (5 ˆ ref ref φ = φ ( v x ( v z (6 and equalty constrants = =,, Z = = ˆ and ˆ = φ + φ = (7 where and are the transformed mole fracton of component at lqud and vapor phase at equlbrum, Z s the global transformed composton of component n the feed, K s the phase equlbrum constant of component, φˆ s the transformed amount fracton for vapor phase whle φ s the conventonal mole fracton of vapor phase whose feasble doman s (,. In ths study, the algorthm used for phase equlbrum calculatons s based on an alternatve Rachford-Rce formulaton for reactve systems usng also transformed varables [5]. Thus, for the reactve flash problem we have ˆ ( = [( ( ( ˆ f φ Z K θ + δ φ ( K θ + ] = (8 = Equaton (8 s an mplct functon used to evaluate φˆ and determne the vapor-lqud equlbrum compostons subject to chemcal equlbrum. We restrct our analyss of reactve flash equatons on the appearance of multple solutons n the regon of physcal sgnfcance for flash separatons,.e. phase equlbrum calculatons are bounded between reactve bubble and dew pont condtons. Usng transformed varables, the reactve flash problem has c r + 2 degrees of freedom. For our analyss, they are fxed by specfyng c r transformed mole fractons Z of the feed, the pressure P of the reactve flash separaton process and a product composton ( or. If under these condtons the system shows MSS, more than one soluton may exst for the reactve flash problem. Monroy-operena [] and Tscareño et al. [4] have suggested that f steady-sate nput multplctes exst n non-reactve mxtures, the correspondng flash equaton for lqud or vapor mole fracton of component must be a concave or convex functon wth respect to temperature or vapor phase fracton. Intutvely, we can expect the same performance for Eq. (2 or (3 n reactve systems usng. Therefore, analyzng the reactve flash problem n the temperature doman between the bubblepont temperature T bub and the dew-pont temperature T dew, whch mples a real vaporlqud equlbrum soluton, and assumng that a maxmum or a mnmum exsts for j T, the correspondng statonary pont s gven by d j dt = (9

5 688 J.E. Jame-eal et al. or, equvalently, for j φˆ we have j ˆ d dφ = ( where j s the lqud ( or vapor phase (. Usng the same analogy reported for nonreactve systems, f the statonary pont s bounded by T [T bub, T dew ] and φ [, ], Eqs. (9 and ( are necessary condtons to dentfy steady-state nput multplctes for the transformed composton varable n ether phase. ote that these condtons are useful to predct the presence of MSS n reactve systems but not for dentfyng the component that shows the multplcty behavor. However, due to the transformaton procedure x, we can easly recognze what component or set of components exhbts steady-state nput multplcty at the specfed operatng condtons. To explore the presence of nput multplcty, the dervatve d j j / dt or d / dφˆ must be evaluated at both the bubble and dew ponts; a change of sgn wll ndcate the exstence of multple solutons for reactve flash problem. These condtons are equvalent to that reported by Monroy-operena [] and Tscareño et al. [4] for non-reactve flash problem. It s convenent to remark that n the general case wthout any smplfcaton of the model used for calculaton of the thermodynamc propertes; these dervatves are evaluated usng fnte dfferences. Alternatvely, a one-dmensonal drect optmzaton strategy can be used to fnd the mnmum (.e. statonary pont for multplcty analyss 2 of d where d s gven by Eq. (9 or (. If the mnmum of d 2 =, the reactve system under analyss shows MSS. Our numercal experence ndcates that an optmzaton approach s more effectve than root-fndng methods for the locaton of the statonary pont of j T and j ˆ φ. 3. umercal example: MSS n methyl tert-butyl ether (MTBE producton We have performed several numercal calculatons on a varety of reactve systems to valdate and verfy the condtons proposed n ths paper for MSS analyss. For llustraton, the MTBE producton s selected as case of study. MTBE s consdered an mportant ndustral chemcal because large quanttes of MTBE have been used as an octane booster n gasolne. Several smulaton and expermental researches have been performed to study the multplcty of MTBE reactve dstllaton process. For the sngle reacton of MTBE producton, we have three reactve components: sobutene ( + methanol (2 methyl tert-butyl ether (3, and one nert component: n-butane (4. In our analyss transformed varables are defned usng MTBE as reference component and are gven by: = (x + x 3 /( + x 3, 2 = (x 2 + x 3 /( + x 3 and 4 = x 4 /( + x 3 where all (,. The reacton takes place n both lqud and vapor phases and thermodynamc propertes are calculated usng Wlson soluton model and Antone equaton wth model parameters reported by Bonlla-Petrcolet et al. [5]. The steady state solutons are found by solvng Eqs. (2 - (8 for an arbtrary transformed feed of Z (.2257,.796,.6947 at dfferent condtons of P. Fgures a and b show the dependence of MSS for and 4 wth respect to the transformed vaporzaton fracton and the temperature. The phase equlbrum behavor of and 4 ndcates the presence of MSS n only the vapor phase for the tested range of. to 22. atm. As expected, Eqs. (9 and ( have changes of sgn when T = T bub and T = T dew ; shows a change of sgn for the statonary condton from. to 22. atm, whle the crteron of nput MSS for 4 s satsfed from. to 2. atm.

6 Predcton of steady-state nput multplctes for reactve flash separaton 689 a φˆ b φˆ Fgure. apor phase composton behavor of (a and (b 4 as a functon of temperature and vaporzaton rato for MTBE producton. By performng the transformaton x, we can easly establsh that sobutene and n- butane have MSS n the vapor phase but at dfferent condtons of P (see Fgure 2. Specfcally, t s observed that the vapor mole fracton of n-butane reaches a maxmum wthn the range of. to 6. atm, whle sobutene shows MSS from. to 22. atm. Tscareño et al. [4] have suggested that the vapor mole fracton of component can show a statonary pont only f K >, whch s consdered a necessary condton for MSS n non-reactve systems. By analyzng the values of K for both sobutene and n- butane (not reported n ths paper, our results ndcate that ths condton s satsfed for the tested range of pressure. Ths agreement suggests that the necessary condtons for MSS proposed for non-reactve systems could be useful to explan the presence of nput multplctes n reactve flash separaton. 4. Conclusons Ths study ntroduces new condtons to dentfy steady-states nput multplctes n reactve flash separaton process, whch are based on the applcaton of reactonnvarant composton varables. Our results ndcate that proposed multplcty analyss

7 69 J.E. Jame-eal et al. s easy to use and effectve for determnng the presence of steady-state nput multplctes n reactve flash operaton. Besdes, our strategy can be appled wth any thermodynamc model and seems sutable for the analyss of mult-reactve and multcomponent systems. Further work wll be focused on the study of MSS n knetcally controlled reactve systems. a.2.9 x φ b x φ Fgure 2. apor phase composton behavor of (a x and (b x 4 as a functon of temperature and vaporzaton rato for MTBE producton. References [] R. Monroy-operena, 2, A fast method to check steady-state nput multplctes n vaporlqud flash separaton, Industral Engneerng Chemstry Research, 4, [2] B.P. Sngh et al., 25, Steady state analyss of reactve dstllaton usng homotopy contnuaton, Chemcal Engneerng Research and Desgn, 83, [3] S. Ung, M.F. Doherty, 995, Theory of phase equlbrum n multreacton systems, Chemcal Engneerng Scence, 5, [4] F. Tscareño et al., 998, Multplcty of the solutons of the flash equatons, Chemcal Engneerng Scence, 53, [5] A. Bonlla-Petrcolet et al., 26, Bubble and dew pont calculatons n multcomponent and multreactve systems, Chemcal Bochemcal Engneerng Quarterly, 2, -8.